Group Theory, Volume 2 |
From inside the book
Results 1-3 of 93
Page 67
Michio Suzuki. Special p - Groups , Extraspecial p - Groups We will first determine the isomorphism classes of p - groups of order p3 . ( 4.13 ) A nonabelian p - group G of order p3 is isomorphic to one of the following groups . If p = 2 ...
Michio Suzuki. Special p - Groups , Extraspecial p - Groups We will first determine the isomorphism classes of p - groups of order p3 . ( 4.13 ) A nonabelian p - group G of order p3 is isomorphic to one of the following groups . If p = 2 ...
Page 73
Michio Suzuki. Proof . Since ( P ) is a proper characteristic subgroup of P , Þ ( P ) is Q- invariant and [ Þ ( P ) , T ] = { 1 } . By ( 1.17 ) , ( P ) is contained in Z ( P ) . The containment P ' ( P ) holds for any p - group P. Let x ...
Michio Suzuki. Proof . Since ( P ) is a proper characteristic subgroup of P , Þ ( P ) is Q- invariant and [ Þ ( P ) , T ] = { 1 } . By ( 1.17 ) , ( P ) is contained in Z ( P ) . The containment P ' ( P ) holds for any p - group P. Let x ...
Page 185
... P- subgroups . We will need the following simple property of P - groups . Assume that the property P satisfies conditions ( 1 ) and ( 3 ) . If a prime q divides the order of a P - group G , then any q - group is a P - group . We define a ...
... P- subgroups . We will need the following simple property of P - groups . Assume that the property P satisfies conditions ( 1 ) and ( 3 ) . If a prime q divides the order of a P - group G , then any q - group is a P - group . We define a ...
Common terms and phrases
A-invariant abelian group abelian normal subgroup abelian subgroup Assume assumption automorphism B₁ central product CG(t CG(x Chapter characteristic subgroup Clearly commutator subgroup conjugacy classes conjugate contradicts coset cyclic group defined definition derived group direct product element g element of order elementary abelian extraspecial factor group finite group following propositions Frobenius functor fusion G contains g of G G satisfies G₁ group G group of order H₁ Hence holds homomorphism implies inductive hypothesis integer involution irreducible characters irreducible representation isomorphic lemma Let G Let H matrix maximal subgroup minimal normal subgroup NG(P nilpotent group nonabelian nontrivial odd order p-group p-nilpotent p-solvable P₁ prime number proof of Theorem proper subgroup Prove the following Q₁ quaternion group S-subgroup satisfies condition shows simple group solvable group subgroup H subgroup of G subgroup of order subset subspace Suppose Sylow U₁ V₁ X₁